Key idea.

Add a constant to create a product of shifted variables.

Rearrange the equation to get \(mn-4m-5n=9\). Adding \(20\) gives

As \(29\) is prime and both factors are integers, the possible values of \(m-5\) are \(1,29,-1,-29\). Therefore

The greatest is \(34\). It is attained when \(n-4=1\), so \(n=5\). Indeed, \(4\cdot34+5\cdot5=161=34\cdot5-9\).